Finite Group Databasebeta

← #29722544

Subgroups — #29722544

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840038C211yesmaximal, characteristic, complemented
2#21455449C212yesmaximal
3C2×#16840003C211yesmaximal, characteristic, complemented
4#21454513C211yesmaximal, characteristic
5#21447181C212yesmaximal, complemented
6#21447120C212yesmaximal, complemented
7#21446732C211yesmaximal, characteristic, complemented
8#21446605C211yesmaximal, characteristic, complemented
9#21446287C212yesmaximal, complemented
10#21445880C211yesmaximal, characteristic, complemented
11C22×#108×C61C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C22×Dic244C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840273C2211yescharacteristic
16C2×#14519756C2212yes—
17#16840038—24no—
18#16840020—22no—
19#16840008—22no—
20#16840003—22no—
21#16840003—21no—
22#16840003—21no—
23#16840003—21no—
24#16840003—21no—
25#16839991—22no—
26#16839833—21no—
27#16839833—21no—
28C8×C22⋊D61—21no—
29C8×C22⋊D61—21no—
30#16839804C2212yes—
31C2×#14519711C2211yescharacteristic
32#16839788C2211yescharacteristic
33#16839772C2212yes—
34#16839764C2212yes—
35C2×#14519702C2212yes—
36C2×#14519695C2212yes—
37C2×#14519695C2212yes—
38#16839728C2211yescharacteristic
39#16839726C2211yescharacteristic
40#16839717C2211yescharacteristic
41#16839530C2212yes—
42#16839530C2212yes—
43C22×M16×C61C2212yes—
44C23×C8×C61C2211yescharacteristic
45C2×#108×C61—24no—
46C2×#108×C61C2212yes—
47C2×#108×C61C2211yescharacteristic
48C2×#108×C61C2211yescharacteristic
49C2×#11758222C2311yescharacteristic
50C22×Dic122C2312yes—
51#14519810—24no—
52#14519810—42no—
53C2×Dic244—22no—
54C2×Dic244—21no—
55C2×Dic244—21no—
56C2×Dic244—22no—
57C2×Dic244—22no—
58C2×#11758190—22no—
59#14519783—22no—
60#14519783—22no—
61C2×#11758194C2311yescharacteristic
62C2×#11758194C2311yescharacteristic
63C2×#11758189—21no—
64C2×C8×D61—21no—
65#14519760—21no—
66#14519756—22no—
67#14519756—22no—
68#14519756—22no—
69#14519756—22no—
70#14519756D411yescharacteristic
71#14519756D411yescharacteristic
72#14519751D411yescharacteristic
73#14519751D411yescharacteristic
74#14519746—21no—
75C4×C22⋊D61—21no—
76C4×C22⋊D61—21no—
77#14519736C2311yescharacteristic
78C2×C4⋊Dic61C2312yes—
79C2×#11758196—22no—
80C2×C22⋊Dic61—22no—
81#14519711—21no—
82#14519711—21no—
83#14519711—22no—
84#14519709—22no—
85#14519706—21no—
86#14519706—21no—
87#14519702—22no—
88#14519702—22no—
89#14519702—22no—
90#14519702—22no—
91#14519702—22no—
92#14519699—22no—
93#14519695—24no—
94#14519695—42no—
95C22⋊#92418—21no—
96C22⋊#92418—21no—
97#14519689—21no—
98#14519689—21no—
99#14519687—21no—
100C8×Dic61—21no—
101C23×C4×C61C2311yescharacteristic
102C2×M16×C61—22no—
103C2×M16×C61—24no—
104C2×M16×C61—42no—
105C2×M16×C61C2312yes—
106C2×M16×C61C2312yes—
107C22×C8×C61—22no—
108C22×C8×C61—22no—
109C22×C8×C61—22no—
110C22×C8×C61D411yescharacteristic
111C22×C8×C61D411yescharacteristic
112C22×C8×C61D411yescharacteristic
113C22×C8×C61D411yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#108×C61—24no—
117#108×C61—22no—
118#108×C61D414yescomplemented
119C24×C61#4111yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—21no—
131C2×C22⋊D61—21no—
132C2×C22⋊D61—21no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135#11758222—21no—
136#11758222—21no—
137#11758222—21no—
138#11758222—21no—
139C2×D244C2×D411yescharacteristic
140C2×C4×D61—21no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122C2×D411yescharacteristic
148M16×C61—22no—
149M16×C61—22no—
150M16×C61—22no—
151M16×C61—44no—
152M16×C61C2×D414yes—
153C2×C8×C61—21no—
154C2×C8×C61—22no—
155C2×C8×C61—21no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—44no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D412yes—
167C2×C8×C61#4112yes—
168C22⋊Dic61—21no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173#11758198—21no—
174#11758198—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D411yescharacteristic
178C4⋊Dic61C2×D411yescharacteristic
179#11758196—21no—
180#11758196—21no—
181#11758196—42no—
182#11758196—42no—
183C4×Dic61—21no—
184#11758194—21no—
185#11758194—21no—
186#11758194—21no—
187#11758194—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190Dic244—44no—
191Dic244—42no—
192Dic244—42no—
193#11758190—44no—
194#11758189—42no—
195C8×D61—42no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61#12111yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61C2×#4111yescharacteristic
203C22×D61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C22⋊D61—42no—
223C22⋊D61—41no—
224C22⋊D61—41no—
225C22⋊D61—42no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—82no—
234D244—21no—
235D244—21no—
236C4×D61—41no—
237C4×D61—41no—
238Dic122—21no—
239Dic122—21no—
240Dic122—42no—
241Dic122—42no—
242Dic122—42no—
243Dic122—42no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—24no—
247C8×C61—42no—
248C8×C61#12112yes—
249C8×C61#12112yes—
250#92418—41no—
251#92418—41no—
252#24804—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61#47312yescomplemented
263C22×C61C2×#12111yescharacteristic
264C22×C61#54811yescharacteristic
265C22×C61#55311yescharacteristic
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61C2×#12111yescharacteristic
272C4×C61#53711yescharacteristic
273C4×C61D4211yescharacteristic
274C4×C61#54711yescharacteristic
275Dic61—41no—
276Dic61—41no—
277Dic61—41no—
278Dic61—82no—
279Dic61—82no—
280C2×#473—611no—
281#3056—612no—
282C2×#467—611no—
283#2929—611no—
284C22×#108D6111yescharacteristic, complemented
285#2064—612no—
286#2052—612no—
287#1979—611no—
288#1952—611no—
289#1901—612no—
290#1816—611no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61—22no—
294C2×C61—41no—
295C2×C61#241611yescharacteristic
296C2×C61C2×#47311yescharacteristic
297C2×C61#306711yescharacteristic
298D61—81no—
299C2×#137—612no—
300C22×Q16—611no—
301C2×#131—611no—
302C22×M16C2×D6112yes—
303C23×C8C2×D6111yescharacteristic
304#525—611no—
305C2×C22⋊Q8—612no—
306#473—1224no—
307#470—1222no—
308#468—1222no—
309#467—1222no—
310#467—1221no—
311#467—1221no—
312#467—1221no—
313#467—1221no—
314#465—1222no—
315#435—1221no—
316#435—1221no—
317C8×D4—1221no—
318C8×D4—1221no—
319C2×#108—24no—
320C2×#108C2×D6112yes—
321C2×#108C2×D6111yescharacteristic
322C2×#108C2×D6111yescharacteristic
323#428—612no—
324C2×#107—611no—
325#424—611no—
326#419—612no—
327#417—612no—
328C2×#103—612no—
329C2×#101—612no—
330C2×#101—612no—
331#408—611no—
332#407—611no—
333#405—611no—
334#343—612no—
335#343—612no—
336C61#2480411yescharacteristic, complemented, Sylow
337C2×#41—611no—
338C22×Q8—612no—
339C23×C4C22×D6111yescharacteristic
340#137—1224no—
341#137—2442no—
342C2×Q16—1222no—
343C2×Q16—1221no—
344C2×Q16—1221no—
345C2×Q16—1222no—
346C2×Q16—1222no—
347C2×SD16—1222no—
348#131—1222no—
349#131—1222no—
350C2×M16—22no—
351C2×M16—24no—
352C2×M16—42no—
353C2×M16—611no—
354C2×M16—611no—
355C2×M16—1221no—
356C2×M16C22×D6112yes—
357C2×M16C22×D6112yes—
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—1221no—
362C22×C8D24411yescharacteristic
363C22×C8D24411yescharacteristic
364C22×C8D24411yescharacteristic
365C22×C8D24411yescharacteristic
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368#123—1221no—
369C22⋊Q8—611no—
370C22⋊Q8—611no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375#121—611no—
376#121—611no—
377#120—1221no—
378C4×D4—1221no—
379C4×D4—1221no—
380#117—611no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1222no—
383C2×C22⋊C4—1222no—
384#108—24no—
385#108—22no—
386#108C22⋊D6114yescomplemented
387#107—1222no—
388#107—1221no—
389#107—1221no—
390#106—1222no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393#103—1222no—
394#103—1222no—
395#103—1222no—
396#103—1222no—
397#103—1222no—
398#102—1222no—
399#101—1224no—
400#101—2442no—
401C22⋊C8—1221no—
402C22⋊C8—1221no—
403C22⋊C8—1221no—
404C22⋊C8—1221no—
405#97—1221no—
406C4×C8—1221no—
407C24#1175822611yescharacteristic
408#41—1221no—
409#41—1221no—
410#41—1221no—
411#41—1221no—
412C2×Q8—611no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×D4—611no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4C2×D24411yescharacteristic
431C22×C4C2×D24411yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4#1175822611yescharacteristic
435C22×C4C23×D6111yescharacteristic
436Q16—2444no—
437Q16—2442no—
438Q16—2442no—
439SD16—2444no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—44no—
444M16—1221no—
445M16—1221no—
446M16—1221no—
447M16—1221no—
448M16—2442no—
449M16C2×C22⋊D6114yes—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—44no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—2442no—
462C2×C8C2×D24411yescharacteristic
463C2×C8C2×D24411yescharacteristic
464C2×C8C2×D24411yescharacteristic
465C2×C8C2×D24411yescharacteristic
466C2×C8#1175822212yes—
467C2×C8C2×C22⋊D6112yes—
468C4⋊C4—611no—
469C4⋊C4—611no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C22⋊C4—1221no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C42—1221no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—41no—
488C23—1221no—
489C23#1451975211yescharacteristic
490C23#1451975211yescharacteristic
491C23C2×#1175822611yescharacteristic
492Q8—1221no—
493Q8—1221no—
494Q8—2442no—
495Q8—2442no—
496Q8—2442no—
497Q8—2442no—
498D4—1221no—
499D4—1221no—
500D4—2442no—
501D4—2441no—
502D4—2442no—
503D4—2441no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—4882no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4#1451975211yescharacteristic
525C2×C4#1451975211yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C22×D24411yescharacteristic
529C2×C4C2×#1175822211yescharacteristic
530C2×C4#1451982211yescharacteristic
531C2×C4#1451982511yescharacteristic
532C8—21no—
533C8—21no—
534C8—24no—
535C8—42no—
536C8—2441no—
537C8—2441no—
538C8#1451975112yes—
539C8#1451975112yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—42no—
548C22—42no—
549C22—2441no—
550C22—2441no—
551C22—2441no—
552C22#1684003812yescomplemented
553C22C2×#1451975211yescharacteristic
554C22#1684042311yescharacteristic
555C22#1684046011yescharacteristic
556C4—21no—
557C4—21no—
558C4—4882no—
559C4—4882no—
560C4—2441no—
561C4—2441no—
562C4—2441no—
563C4C2×#1451975111yescharacteristic
564C4#1684034311yescharacteristic
565C4D4×C22⋊D6111yescharacteristic
566C4#1684041311yescharacteristic
567C2—21no—
568C2—22no—
569C2—21no—
570C2—41no—
571C2—4881no—
572C2#2144925311yescharacteristic
573C2C2×#1684003811yescharacteristic
574C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#108×C61C2
socleC22×C61C2×#121
prime radical#16840273C22
hypercenterC22×#108D61
hypocenterC61#24804
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840038C2semidirect
3. C2×#16840003C2semidirect
5. #21447181C2semidirect
6. #21447120C2semidirect
7. #21446732C2semidirect
8. #21446605C2semidirect
9. #21446287C2semidirect
10. #21445880C2semidirect
11. C22×#108×C61C2semidirect
118. #108×C61D4semidirect
262. C22×C61#473semidirect
284. C22×#108D61semidirect
336. C61#24804semidirect
386. #108C22⋊D61semidirect
552. C22#16840038semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400382
#214554492
C2×#168400032
#214545132
#214471812
#214471202
#214467322
#214466052
#214462872
#214458802
C22×#108×C612
#2480461